Opinions on a subject with just two sides can nonetheless be held with different degrees of conviction, which leads to different patterns of influence when individuals interact. Individuals with more strongly-held opinions can have greater influence than those with moderate opinions, or less, if others are wary of extreme views. We use a differential equation model to understand how the distribution of opinions evolves under various interaction and influence scenarios. In particular, we explore the conditions that lead to polarisation (the division of a population into two camps each holding only the corresponding extreme view) and centering (a population largely distributed on more moderate opinions on both sides of an issue).